Given a snooth projective variety \(X\) of dimension \(n\) and an embedding \(i:X\hookrightarrow \mathbb P^N\), a vector bundle \(E\) on \(X\) is called Steiner if it is contained in the short exact sequence of vector bundles \[ 0\to i^* \mathcal O_{\mathbb P^N}(-1)^{\oplus s}\to i^* \mathcal O_{\mathbb P^N}^{\oplus t}\to E\to 0 \] for some integer \(s,t>0\). C. Brambilla [Math. Nachr. 281, No. 4, 499–516 (2008; Zbl 1156.14013)] proved that a generic Steiner bundle \(E\) on \(\mathbb P^n\) is exceptional if and only if \(\chi(\text{End} E)=1\). Moreover, it is proven that any exceptional Steiner bundle on \(\mathbb P^n\) is stable for any \(n\geq 2\). In the paper under review, the author extends these results to any generic Steiner bundle \(E\) on a smooth hyperquadric \(Q_n\subseteq \mathbb P^{n+1}\), for any \(n\geq 3\).